Write an Equation When Given Its Solutions

Vieta's formulas

Write a Quadratic Equation When Given Its Solutions

If ${\displaystyle x_1}$ and ${\displaystyle x_2}$ are the roots of the quadratic equation ${\displaystyle {\mathrm{ax}}^2+\mathrm{<mn>b</mn>}x+\mathrm{<mn>c</mn>}}$ = ${\displaystyle 0}$ then:
$x_1+x_2$ = ${\displaystyle -\frac{b}{a}}$
${\displaystyle x_1\hspace{0.17em}·\hspace{0.17em}{x}_2}$ = ${\displaystyle \frac{c}{a}}$

If ${\displaystyle x_1}$ and ${\displaystyle x_2}$ are the roots of the quadratic equation ${\displaystyle x^2+px+q}$ = ${\displaystyle 0}$ then:
$x_1+x_2$ = $-p$
${\displaystyle x_1\hspace{0.17em}·\hspace{0.17em}{x}_2}$ = $q$

Write a Cubic Equation When Given Its Solutions

If ${\displaystyle x_1}$, ${\displaystyle x_2}$ and ${\displaystyle x_3}$ are the roots of the cubic equation ${\displaystyle {\displaystyle a{x}^3+b{x}^2+cx+d}}$ = ${\displaystyle 0}$ then:
${\displaystyle x_1+x_2+x_3}$ = ${\displaystyle -\frac{b}{a}}$
${\displaystyle x_1{x}_2+x_1{x}_3+x_2{x}_3}$ = ${\displaystyle \frac{c}{a}}$
${\displaystyle x_1\hspace{0.17em}·\hspace{0.17em}{x}_2\hspace{0.17em}·\hspace{0.17em}{x}_3}$ = ${\displaystyle -\frac{d}{a}}$